

A287650


Number of doubly symmetric diagonal Latin squares of order 4n with a fixed first row.


5




OFFSET

1,1


COMMENTS

A doubly symmetric square has symmetries in both the horizontal and vertical planes.
One plane symmetry requires onetoone correspondence between the values of elements a[i][j] and a[Ni][j] in a vertical plane, and between the values of elements a[i][j] and a[i][Nj] in a horizontal plane for 1 <= i,j <= N.  Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).


LINKS

Table of n, a(n) for n=1..3.
A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, value a(4) is wrong (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98100 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 7479.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 1719 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 3036 (in Russian)
Index entries for sequences related to Latin squares and rectangles


FORMULA

a(n) = A292517(n) / (4n)!.


EXAMPLE

Doubly symmetric diagonal Latin square example:
0 1 2 3 4 5 6 7
3 2 7 6 1 0 5 4
2 3 1 0 7 6 4 5
6 7 5 4 3 2 0 1
7 6 3 2 5 4 1 0
4 5 0 1 6 7 2 3
5 4 6 7 0 1 3 2
1 0 4 5 2 3 7 6
Reversion of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reversion of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric.


CROSSREFS

Cf. A003191, A287649, A292517.
Sequence in context: A134656 A128122 A082178 * A082912 A265013 A083973
Adjacent sequences: A287647 A287648 A287649 * A287651 A287652 A287653


KEYWORD

bref,nonn,more


AUTHOR

Eduard I. Vatutin, May 29 2017


EXTENSIONS

a(2) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Edited and a(3) from Alexey D. Belyshev added by Max Alekseyev, Aug 23 2018, Sep 07 2018


STATUS

approved



